Double Higgs production at the linear colliders and the probing of the Higgs self-coupling

We study double Higgs production in the e⁺e^? and γγ modes of the linear collider. It is also shown how one can probe the scalar potential in these reactions. We discuss the effective longitudinal W approximation in γγ processes and the W _LW_L luminosities in the two modes of a high-energy linear collider. A generalised non-linear gauge-fixing condition, which is particularly useful for tree-level calculations of electroweak processes for the laser induced collider, is presented. Its connection with the background-field approach to gauge fixing is given.